From Wikipedia, the free encyclopedia

In mathematics, a **syzygy** (from Greek συζυγία ‘pair’) is a relation between the generators of a module *M*. The set of all such relations is called the “first syzygy module of *M*“. A relation between generators of the first syzygy module is called a “second syzygy” of *M*, and the set of all such relations is called the “second syzygy module of *M*“. Continuing in this way, we derive the *n*th syzygy module of *M* by taking the set of all relations between generators of the (*n* − 1)^{th} syzygy module of *M*. If *M* is finitely generated over a polynomial ring over a field, this process terminates after a finite number of steps; i.e., eventually there will be no more syzygies (see Hilbert’s syzygy theorem). The syzygy modules of *M* are not unique, for they depend on the choice of generators at each step.

The sequence of the successive syzygy modules of a module *M* is the sequence of the successive images (or kernels) in a free resolution of this module.

Buchberger’s algorithm for computing Gröbner bases allows the computation of the first syzygy module: The reduction to zero of the S-polynomial of a pair of polynomials in a Gröbner basis provides a syzygy, and these syzygies generate the first module of syzygies.

## Further reading